Bad configuration, possible bug?
Blender uses a user configurable shortcut system. We can easily add a shortcut to many operators by  on the button or menu item and choosing Add Shortcut. If you then managed to right click on the shortcut entry dialog you would get the right click as the shortcut.



If this is what you did, you will see Right Mouse next to the plane in the add menu &#8679; ShiftA->Mesh.



To remove it should be as simple as quitting blender and starting it again. If however you saved the user settings after adding the shortcut, then you will need to remove it manually or revert to factory settings. Note that adding a keyboard shortcut can be removed by right clicking it again, but a mouse shortcut needs to be removed in the input settings.



In User Preferences under the Input tab you will find the shortcut. For adding an object, you will find it by expanding 3D View and Object Mode and scrolling down to the bottom of that list.



Click the X icon to the right of the entry to remove it.



In your case you could also search for plane to find it, other times searching may be the easiest way to find it, if you can think of the right search words. You can also search by shortcut, but you will find many entries for each key and mouse so it doesn't always help.



Be sure to save user settings after removing the shortcut so that it doesn't return next time you use blender.